Exact measurement of the properties of nature is a common goal within the experimental sciences. Similarly, medical diagnostics and communications technology, among other scientific endeavors, seek the ability to obtain exact measurement of properties within their respective fields, e.g., MRI or free space optical transmission. However, in spite of the availability of highly sophisticated instruments, instrumental signatures are present in the data, making the measurement only approximate. An area of experimental science in which instrumental signatures are particularly a problem is astronomy, where the sources to be measured are very faint. Even when the instruments are made essentially noise-free, instrumental signatures related to finite spatial, spectral, or temporal resolution remain. At this point, image reconstruction is required to remove the instrumental signatures.
One of the most powerful approaches to image restoration, e.g., removal of point-spread-function blurring, is Bayesian image reconstruction, which includes goodness-of-fit and maximum entropy. This family of techniques employs a statistical relationship between various quantities involved in the imaging process. Specifically, the data, D, consisting of the original noisy, blurred image is linked to the noise-free, unblurred image, I, through a model, M. The model M includes all aspects of the relationship between the data and the image, e.g., that the data is normally collected on a rectangular grid and that the data is related to the image through the relationship EQU D(i)=(I*H)(i)+N(i), (1)
where D(i) is the data in cell i (typically a pixel), I is the image, H is the point-spread-function (PSF), * is the spatial convolution operator, i.e., ##EQU1## and N represents the noise in the data.
To statistically model the imaging process, the properties of the joint probability distribution of the triplet, D, I and M, i.e., p(D,I,M), are analyzed. Applying Bayes' Theorem p(A,B)=p(A.vertline.B)p(B)=p(BIA)p(A), where p(X.vertline.Y) is the probability of X given that Y is known) provides: ##EQU2## By setting the first factorization of p(D,I,M) in equation 3 equal to the second factorization provides the usual starting point for Bayesian reconstruction: ##EQU3## The goal of Bayesian image reconstruction is to find the M.A.P. (Maximum A Posteriori) image, I, which maximizes p(I.vertline.D,M), i.e., the most probable image given the data and model. (Note that other image estimates, e.g., the average image, &lt;I&gt;=.intg..sub.D,M dMdD I p(I.vertline.D,M), may be used here and in the methods described in the detailed description.)
It is common in Bayesian image reconstruction to assume that the model is fixed. In this case, p(D.vertline.M) is constant, so that EQU p(I.vertline.D,M).varies.p(D.vertline.I,M)p(I.vertline.M). (5)
The first term, p(D.vertline.I,M), is a goodness-of-fit quantity, measuring the likelihood of the data given a particular image and model. The second term, p(I.vertline.M), is normally referred to as the "image prior", and expresses the a priori probability of a particular realization of the image given the model. In goodness-of-fit (GOF) image reconstruction, p(I.vertline.M) is effectively set to unity, i.e., there is no prior bias concerning the image. Only the goodness-of-fit (p(D.vertline.I,M)) is maximized during image reconstruction. Typically, EQU p(I.vertline.D,M)=p.sub..chi..spsb.2 (.chi..sup.2.sub.R), (6)
where .chi..sup.2.sub.R is the chi-square of the residuals, R (.ident.D-I*H), and p.sub..chi..spsb.2 is the .chi..sup.2.sbsp.- distribution. While this approach ensures that the frequency distribution of the residuals has a width which is characteristic of the noise distribution, it normally results in images with spurious spatial features where the data has a low signal to noise ratio (SNR). Also, the large amplitude residuals often show a strong spatial correlation with bright features in the data.
Maximum entropy (ME) image reconstruction solves many of the problems of the simpler GOF methods. In ME imaging, one calculates a value for the image prior based upon "phase space volume" or counting arguments. Heuristically, p(I.vertline.M) is written p(I.vertline.M)=exp(S), where S is the entropy of the image in a given model. All ME methods capitalize on the virtues of incorporating prior knowledge of the likelihood of the image. The benefits of this are numerous, including eliminating the over-resolution problems of GOF methods and increasing the numerical stability of the calculations.
Many Bayesian image reconstruction methods assume that the model is fixed. However, recent advances in ME reconstruction propose varying the model. A significant development in this area is the multi-channel image modeling of Weir (Applications of Maximum Entropy Techniques to HST Data, Proceedings of the ESO/ST-ECF Data Analysis Workshop, April 1991). In this method the image is assumed to be a sum of pseudo-images convolved with a blurring function of various spatial scales. This method, while superior to many of its predecessors, may exhibit low-level spurious sources as ripples in the reconstructed image, and still displays some spatial correlation within the residuals.